Problem: A cube has edges of length 1 cm and has a dot marked in the centre of the top face.  The cube is sitting on a flat table.  The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table.  The cube is rolled until the dot is again on the top face.  The length, in centimeters, of the path followed by the dot is $c\pi$, where $c$ is a constant.  What is $c$?
Suppose the cube rolls first over edge $AB$.

Consider the cube as being made up of two half-cubes (each of dimensions $1 \times 1 \times \frac{1}{2}$) glued together at square $PQMN$.  (Note that $PQMN$ lies on a vertical plane.)

Since dot $D$ is in the centre of the top face, then $D$ lies on square $PQMN$. [asy]
//C24S4

size(4cm);

pair shiftpair = 0.3 * (-Sin(50), Sin(40));

// Draw squares
draw(unitsquare);
draw(shift(shiftpair) * unitsquare);
draw(shift(2 * shiftpair) * unitsquare);

// Draw lines
pair[] vertices = {(0, 0), (1, 0), (1, 1), (0, 1)};
int i;
for (i = 0; i < 4; ++i) {

pair inner = vertices[i];

pair outer = shift(2 * shiftpair) * inner;

draw(inner--outer);
}

// Point labels
label("$A$", (1, 0), SE);
label("$B$", shift(2 * shiftpair) * (1, 0), NW);
picture pic;
label(pic, "$N$", (0, 0), SW);
label(pic, "$M$", (1, 0), NE);
label(pic, "$Q$", (1, 1), NE);
label(pic, "$D$", (0.5, 1), N); dot(pic, (0.5, 1));
label(pic, "$P$", (0, 1), NE);
add(shift(shiftpair) * pic);
[/asy] Since the cube always rolls in a direction perpendicular to $AB$, then the dot will always roll in the plane of square $PQMN$. [asy]
//C24S1
size(2.5cm);
draw(unitsquare);
label("$N$", (0, 0), SW);
label("$M$", (1, 0), SE);
label("$Q$", (1, 1), NE);
label("$D$", (0.5, 1), N); dot((0.5, 1));
label("$P$", (0, 1), NW);
[/asy] So we can convert the original three-dimensional problem to a two-dimensional problem of this square slice rolling.

Square $MNPQ$ has side length 1 and $DQ=\frac{1}{2}$, since $D$ was in the centre of the top face.

By the Pythagorean Theorem, $MD^2 = DQ^2 + QM^2 = \frac{1}{4}+1= \frac{5}{4}$, so $MD = \frac{\sqrt{5}}{2}$ since $MD>0$. In the first segment of the roll, we start with $NM$ on the table and roll, keeping $M$ stationary, until $Q$ lands on the table. [asy]
//C24S2
size(4cm); // ADJUST

// Draw outline
draw(unitsquare);
draw((0, 0)--(-1, 0)--(-1, 1)--(0, 1), dashed);
draw((-0.5, 1)--(0, 0)--(1, 0.5), dashed);

// Labels and dots
label("$N$", (0, 1), SE);
label("$M$", (0, 0), S);
label("$Q$", (1, 0), SE);
label("$D$", (1, 0.5), E); dot((1, 0.5));
label("$P$", (1, 1), NE);
dot((-0.5, 1));

// Draw arc
draw(reverse(arc((0, 0), (1, 0.5), (-0.5, 1))), dashed, MidArcArrow(size=6));
[/asy] This is a rotation of $90^\circ$ around $M$.  Since $D$ is at a constant distance of $\frac{\sqrt{5}}{2}$ from $M$, then $D$ rotates along one-quarter (since $90^\circ$ is $\frac{1}{4}$ of $360^\circ$) of a circle of radius $\frac{\sqrt{5}}{2}$, for a distance of $\frac{1}{4}\left( 2\pi\frac{\sqrt{5}}{2}\right) = \frac{\sqrt{5}}{4}\pi$.

In the next segment of the roll, $Q$ stays stationary and the square rolls until $P$ touches the table.  [asy]
//C24S3

size(4cm); // ADJUST

// Draw outline
draw(unitsquare);
draw((0, 0)--(-1, 0)--(-1, 1)--(0, 1), dashed);
draw((-1, 0)--(-2, 0)--(-2, 1)--(-1, 1), dashed);

// Labels and dots
dot((-1.5, 1));
label("$M$", (0, 1), N);
label("$Q$", (0, 0), S);
label("$P$", (1, 0), SE);
label("$D$", (0.5, 0), S); dot((0.5, 0));
label("$N$", (1, 1), NE);
dot((0, 0.5));

// Draw arc
draw(reverse(arc((0, 0), (0.5, 0), (0, 0.5))), dashed, MidArcArrow(size=6));
[/asy] Again, the roll is one of $90^\circ$.  Note that $QD = \frac{1}{2}$.  Thus, again $D$ moves through one-quarter of a circle this time of radius $\frac{1}{2}$,  for a distance of $\frac{1}{4}\left( 2\pi \frac{1}{2}\right) =\frac{1}{4}\pi$.

Through the next segment of the roll, $P$ stays stationary and the square rolls until $N$ touches the table.  This is similar to the second segment, so $D$ rolls through a distance of $\frac{1}{4}\pi$.

Through the next segment of the roll, $N$ stays stationary and the square rolls until $M$ touches the table.  This will be the end of the process as the square will end up in its initial position.  This segment is similar to the first segment so $D$ rolls through a distance of $\frac{\sqrt{5}}{4}\pi$.

Therefore, the total distance through which the dot travels is  $$\frac{\sqrt{5}}{4}\pi+\frac{1}{4}\pi+\frac{1}{4}\pi+\frac{\sqrt{5}}{4}\pi$$or $$\left(\frac{1+\sqrt{5}}{2}\right)\pi,$$so our final answer is $\boxed{\dfrac{1+\sqrt{5}}{2}}$.